Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions

In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:msub><a:mrow><a:mi mathvariant="double-struck">Z</a:mi></a:mrow><a:mi>d</a:mi></a:msub></a:math> qudits, including instances where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><e:mi>d</e:mi></e:math> is a nonprime number. This capability allows the identification of topological orders that differ from the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:msub><h:mrow><h:mi mathvariant="double-struck">Z</h:mi></h:mrow><h:mi>d</h:mi></h:msub></h:math> toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><l:msub><l:mrow><l:mi mathvariant="double-struck">Z</l:mi></l:mrow><l:mi>p</l:mi></l:msub></l:math> qudits (with <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><p:mi>p</p:mi></p:math> being a prime) are equivalent to finite copies of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><s:msub><s:mrow><s:mi mathvariant="double-struck">Z</s:mi></s:mrow><s:mi>p</s:mi></s:msub></s:math> toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order.